Question

How to calculate the volume and surface area of a cone

Answer 1

To calculate the volume and surface area of a cone, you can use the following formulas:

### 1. Volume of a Cone
The volume \( V \) of a cone can be calculated using the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

Where:
- \( V \) = volume of the cone
- \( r \) = radius of the base of the cone
- \( h \) = height of the cone
- \( \pi \) (pi) is approximately 3.14159.

### 2. Surface Area of a Cone
The surface area \( A \) of a cone is the sum of the base area and the lateral (side) area. The formula is:

\[
A = \pi r^2 + \pi r l
\]

Where:
- \( A \) = surface area of the cone
- The first term \( \pi r^2 \) is the area of the base of the cone.
- The second term \( \pi r l \) is the lateral surface area.
- \( l \) = slant height of the cone, which can be calculated using the Pythagorean theorem if you know the radius and height:

\[
l = \sqrt{r^2 + h^2}
\]

### Summary of Steps:
1. **Calculate the slant height \( l \)** if needed:
\[
l = \sqrt{r^2 + h^2}
\]

2. **Calculate the volume \( V \)**:
\[
V = \frac{1}{3} \pi r^2 h
\]

3. **Calculate the surface area \( A \)**:
\[
A = \pi r^2 + \pi r l
\]

### Example
For a cone with a radius of 3 units and a height of 4 units:

1. Calculate the slant height:
\[
l = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ units}
\]

2. Calculate the volume:
\[
V = \frac{1}{3} \pi (3^2)(4) = \frac{1}{3} \pi \cdot 9 \cdot 4 = 12\pi \approx 37.68 \text{ cubic units}
\]

3. Calculate the surface area:
\[
A = \pi (3^2) + \pi (3)(5) = \pi (9 + 15) = 24\pi \approx 75.40 \text{ square units}
\]

This method should help you calculate the volume and surface area of a cone for any given dimensions.